UNICITY THEOREMS CONCERNING A L-FUNCTION AND A MEROMORPHIC FUNCTION

Let h be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if <TEX>$f^m$</TEX> is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type <TEX>$$f^n+P(f)=be^{sz}+Q(e^z)$$</TEX>, where P(f) is a differential polynomial in f of degree at most n-1, and Q(<TEX>$e^z$</TEX>) is a polynomial in <TEX>$e^z$</TEX> of degree k <TEX>$\leqslant$</TEX> max {n-1, s(n-1)/n} with small functions of <TEX>$e^z$</TEX> as its coefficients.

We study the uniqueness of meromorphic functions concerning nonlinear differential polynomials sharing a nonzero polynomial IM.Though the main concern of the paper is to improve a recent result of the present author [12], as a consequence of the main result we also generalize two recent results of X. M. Li and L. Gao [11].

In this paper, we will give suitable conditions on differential polynomials [Formula: see text] such that they take every finite nonzero value infinitely often, where [Formula: see text] is a meromorphic function in complex plane. These results are related to Problems 1.19 and 1.20 in a book of Hayman and Lingham [Research Problems in Function Theory, preprint (2018), https://arxiv.org/pdf/1809.07200.pdf ]. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials [Formula: see text] to have some terms of any degree of [Formula: see text] and also the hypothesis [Formula: see text] in [Theorem 2 of W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11(2) (1995) 355–373] is replaced by [Formula: see text] in our result.

In this paper we discuss the uniqueness problem for differential and difference polynomials of the form (fnm(z)fnd(qz + c))(k) for meromorphic functions in a non-Archimedean field.

We study the uniqueness of meromorphic functions that share one small function with more general differential polynomial P[f]. As corollaries, we obtain results which answer open questions posed by Yu (2003).

In this paper, we study the uniqueness problem of certain differential polynomials generated by two meromorphic functions. The results of the paper extend some recent results due to Meng and Li (J Anal 28:1–6, 2019).

Some applications of the Tumura-Clunie theorem are given.Most of these concerned fixed points of compositions of entire functions.1. Introduction.The purpose of this note is to develop some applications of the Tumura-Clunie theory (Hayman [2, p. 69]).It is assumed that the reader is familiar with the basic quantities of Nevanlinna Theory: T(r,f), m(r,f), N(r,f),... and their elementary properties.II.Notation, terminology.1. /will always denote a nonconstant meromorphic function.2. We shall denote by S(r,f) any quantity satisfying S(r,f) = o{T{r,f)}, as r - + oo, possibly outside a set of r of finite measure.3. We shall denote by a(z)(ax(z),...) meromorphic functions satisfying T(r,a(z)) = S(r,f).4. By a differential polynomial P(f) we understand a polynomial in/(z) and its derivatives whose coefficients are of the form a(z).III. Statement and applications of the Tumura-Clunie theorem.Theorem 1 (Tumura-Clunie[2]).Suppose g(z)=fn(z)+Pn_x(f), and that(1) N(r,f) + N(r, l/g) = S(r,f).Then g(z) = hn(z), h(z)=f(z) + a(z)/n and hn~\z)a{z) is obtained by substituting h(z) for f(z), h'(z) for f'(z), etc. in the terms of degree n-\ in Pn-x(J~).Remark.The conclusion still holds good if the condition (1) is replaced by: N(r,f)+N(r, l/g) = S0(r,f), where S0(r,f) denotes any quantity which satisfies S0(r,f) = o(T(r,f)) as r - +00 through a set of r of infinite measure.We have

Entire or meromorphic functions sharing one value

Elementary proofs of several generalizations of Tse's extension of an asymptotic radial uniqueness theorem of Barth and Schneider are given.Let A = {| z |< 1} and C -{| z |= 1}.The following is an extension to meromorphic functions by Tse [5] of a theorem of Barth and Schneider [1].Theorem 1.Let ll be a positive monotone decreasing function with domain [0,1) such that limr_xp(r) = 0. Let S be a second category subset of C. If f is a meromorphic function on A with the property that f(rr\) = o[n(r)]for each tj G S, then / = 0.Barth and Schneider's proof (for bounded analytic functions) depends on deep theorems of Mergelyan, Lusin-Privalov, and Collingwood.Tse's proof of Theorem 1 is based on the method of Barth and Schneider.An equivalent formulation [4, Corollary 2] of Theorem 1 may be obtained in a more straightforward manner as a corollary of a theorem of Rippon [4, Theorem 1]; however, this proof still relies on the Collingwood maximality theorem as well as results and methods needed for a proof of the Lusin-Privalov theorem.The proofs that we shall give to several generalizations of Theorem 1 are elementary and are based on the following.Category Principle.Let S be a second category set.If S -U"=1 Fn with each F" closed, then some Fn contains a nonempty open set.Proof.Since E is of second category, some Fn must be dense in an open set U. Since Fn is closed, we have U E Fn as required.We turn now to our first generalization of Theorem 1.Let be a continuum contained in A = (| z |< 1} such that D C = {1} and let ,, = {t/z: z G } for each 7) E C. For p a positive function with domain [0,1) such that limrjl p(r) = 0 and tj G C, we shall write/Theorem 2. Let p be a positive function with domain [0,1) such that limr^ x p(r) = 0 and S a second category subset of C. If f is a meromorphic function on A with the property that f(z) = 0[p(\ z |)], z G ,,, for each 17 G S, then / = 0.

PREFACE. INTRODUCTION. Generalities. Ordinary differential equations. Supplementary conditions associated to ODEs. The Cauchy (initial) problem.The two-point problem. 1: LINEAR ODEs OF FIRST AND SECOND ORDER. 1.1 Linear first order ODEs. 1.1.1 Equations of the form . 1.1.2 The linear homogeneous equation. 1.1.3 The general case. 1.1.4 The method of variation of parameters (Lagrange's method). 1.1.5 Differential polynomials. 1.2 Linear second order ODEs. 1.2.1 Homogeneous equations. 1.2.2 Non-homogeneous equations. Lagrange's method. 1.2.3 ODEs with constant coefficients. 1.2.4 Order reduction. 1.2.5 The Cauchy problem. Analytical methods to obtain the solution. 1.2.6 Two-point problems (Picard). 1.2.7 Sturm-Liouville problems. 1.2.8 Linear ODEs of special form. 1.3. Applications 2: LINEAR ODEs OF HIGHER ORDER (n >2). 2.1 The general study of linear ODEs of order . 2.1.1 Generalities. 2.1.2 Linear homogeneous ODEs. 2.1.3 The general solution of the non-homogeneous ODE. 2.1.4 Order reduction. 2.2 Linear ODEs with constant coefficients. 2.2.1 The general solution of the homogeneous equation. 2.2.2 The non-homogeneous ODE. 2.2.3 Euler type ODEs. 2.3 Fundamental solution. Green function. 2.3.1 The fundamental solution. 2.3.2 The Green function. 2.3.3 The non-homogeneous problem. 2.3.4 The homogeneous two-point problem. Eigenvalues. 2.4 Applications. 3: LINEAR ODSs OF FIRST ORDER. 3.1 The general study of linear first order ODSs. 3.1.1 Generalities. 3.1.2 The general solution of the homogeneous ODS. 3.1.3 The general solution of the non-homogeneous ODS. 3.1.4 Order reduction of homogeneous ODSs. 3.1.5 Boundary value problems for ODSs. 3.2 ODSs with constant coefficients. 3.2.1 The general solution of the homogeneous ODS. 3.2.2 Solutions in matrix form for linear ODSs with constant coefficients. 3.3 Applications. 4: NON-LINEAR ODEs OF FIRST AND SECOND ORDER. 4.1 First order non-linear ODEs. 4.1.1 Forms of first order ODEs and oftheir solutions. 4.1.2 Geometric interpretation. The theorem of existence and uniqueness. 4.1.3 Analytic methods for solving first order non-linear ODEs. 4.1.4. First order ODEs integrable by quadratures. 4.2 Non-linear second order ODEs. 4.2.1 Cauchy problems. 4.2.2 Two-point problems. 4.2.3 Order reduction of second order ODEs. 4.2.4 The Bernoulli-Euler equation. 4.2.5 Elliptic integrals. 4.3 Applications. 5: NON-LINEAR ODSs OF FIRST ORDER. 5.1 Generalities. 5.1.1 The general form of a first order ODS. 5.1.2 The existence and uniqueness theorem for the solution of the Cauchy problem. 5.1.3 The particle dynamics. 5.2 First integrals of an ODS. 5.2.1 Generalities. 5.2.2 The theorem of conservation of the kinetic energy. 5.2.3 The symmetric form of an ODS. Integral combinations. 5.2.4 Jacobi's multiplier. The method of the last multiplier. 5.3 Analytical methods of solving the Cauchy problem for non-linear ODSs. 5.3.1 The method of successive approximations (Picard-Lindeloff). 5.3.2 The method of the Taylor series expansion. 5.3.3 The linear equivalence method (LEM). 5.4 Applications. 6: VARIATIONAL CALCULUS. 6.1 Necessary condition of extremum for functionals of integral type. 6.1.1 Generalities. 6.1.2 Functionals of the form..... 6.1.3 Functionals of the form..... 6.1.4 Functionals of integral type, depending on n functions. 6.2 Conditional extrema. 6.2.1 Isoperimetric problems. 6.2.2 Lagrange's problem. 6.3 Applications. 7: STABILITY. 7.1 Lyapunov Stability. 7.1.1 Generalities. 7.1.2 Lyapunov's theorem of stability. 7.2 The stability of the solutions of dynamical systems. 7.2.1 Autonomous dynamical systems. 7.2.2 Long term behaviour of the solutions. 7.3 Applications. INDEX. REFERENCES.

Uniqueness results of meromorphic functions whose derivatives share four small functions

In this paper, we investigate meromorphic functions that share a small function with one of its linear differential polynomials and prove several theorems which generalize and improve the main results given by J. L. Zhang and L. Z. Yang. Some examples are provided to show that the results in this paper are sharp.

In this paper, we study the problem of meromorphic functions that share one small function of differential polynomial with their derivatives and prove one theorem. The theorem improves the results of Jin-Dong Li and Guang-Xin Huang [1].

In this paper we aim to establish some results depending on the comparative growth properties of composite transcendental entire or meromorphic functions and some special type of differential polynomials generated by one of the factors on the basis of generalized order α,β and generalized lower order α,β where α and β are continuous non-negative functions defined on −∞,+∞.

We will study the uniqueness problems of meromorphic functions of differential polynomials sharing fixed points. Our results improve or generalize some previous results on meromorphic functions sharing fixed points.